A010010 -id:A010010 - cp's OEIS frontend

A033571 a(n) = (2n + 1)(5*n + 1).

1, 18, 55, 112, 189, 286, 403, 540, 697, 874, 1071, 1288, 1525, 1782, 2059, 2356, 2673, 3010, 3367, 3744, 4141, 4558, 4995, 5452, 5929, 6426, 6943, 7480, 8037, 8614, 9211, 9828, 10465, 11122, 11799, 12496, 13213, 13950, 14707, 15484, 16281, 17098, 17935, 18792, 19669, 20566, 21483

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This is one of the diagonals in the spiral. - Omar E. Pol, Sep 10 2011

Also sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is a line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Stellar Layers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A153127. - Reinhard Zumkeller, Dec 20 2008
Cf. A000566, A008596, A010010, A153126, A158186.
Cf. A001622, A003154, A007742, A019952.

List([0..50], n-> (2*n+1)*(5*n+1)); # G. C. Greubel, Oct 12 2019

[(2*n+1)*(5*n+1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq((2*n+1)*(5*n+1), n=0..50); # G. C. Greubel, Oct 12 2019

Table[(2*n+1)*(5*n+1), {n,0,50}] (* G. C. Greubel, Oct 12 2019 *)

a(n)=(2*n+1)*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

[(2*n+1)*(5*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

a(n) = A153126(2*n) = A000566(2*n+1). - Reinhard Zumkeller, Dec 20 2008
From Reinhard Zumkeller, Mar 13 2009: (Start)
a(n) = A008596(n) + A158186(n), for n > 0.
a(n) = A010010(n) - A158186(n). (End)
a(n) = a(n-1) + 20*n - 3 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Oct 12 2019: (Start)
G.f.: (1 + 15*x + 4*x^2)/(1-x)^3.
E.g.f.: (1 + 17*x + 10*x^2)*exp(x). (End)
a(n) = A003154(n+1) + A007742(n). - Leo Tavares, Mar 27 2022
Sum_{n>=0} 1/a(n) = sqrt(1+2/sqrt(5))*Pi/6 + sqrt(5)*log(phi)/6 + 5*log(5)/12 - 2*log(2)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022

Terms a(36) onward added by G. C. Greubel, Oct 12 2019

A158187 a(n) = 10*n^2 + 1.

Original entry on oeis.org

1, 11, 41, 91, 161, 251, 361, 491, 641, 811, 1001, 1211, 1441, 1691, 1961, 2251, 2561, 2891, 3241, 3611, 4001, 4411, 4841, 5291, 5761, 6251, 6761, 7291, 7841, 8411, 9001, 9611, 10241, 10891, 11561, 12251, 12961, 13691, 14441, 15211, 16001, 16811, 17641

Offset: 0

Reinhard Zumkeller, Mar 13 2009

Sequence found by reading the segment (1, 11) together with the line from 11, in the direction 11, 41, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

The identity (10n^2 + 1)^2 - (25n^2 + 5)*(2n)^2 = 1 can be written as a(n)^2 - A158445(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Jan 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A158445, A005843. - Vincenzo Librandi, Mar 19 2009

Table[10*n^2+1,{n,0,50}] (* Vincenzo Librandi, Jan 03 2012 *)

a(n)=10*n^2+1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = A033583(n) + 1.
For n > 0: a(n) = A010010(n)/2.
From Vincenzo Librandi, Jan 03 2012: (Start)
G.f: x*(11 + 8*x + x^2)/(1-x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(10))*coth(Pi/sqrt(10)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(10))*csch(Pi/sqrt(10)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(10))*sinh(Pi/sqrt(5)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(10))*csch(Pi/sqrt(10)). (End)
E.g.f.: exp(x)*(1 + 10*x + 10*x^2). - Stefano Spezia, Feb 05 2021

A158186 a(n) = 10n^2 - 7n + 1.

Original entry on oeis.org

1, 4, 27, 70, 133, 216, 319, 442, 585, 748, 931, 1134, 1357, 1600, 1863, 2146, 2449, 2772, 3115, 3478, 3861, 4264, 4687, 5130, 5593, 6076, 6579, 7102, 7645, 8208, 8791, 9394, 10017, 10660, 11323, 12006, 12709, 13432, 14175, 14938, 15721, 16524, 17347

Offset: 0

Reinhard Zumkeller, Mar 13 2009

Sequence found by reading the segment (1, 4) together with the line (one of the diagonal axes) from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A008596, A010010, A033571, A085787.

Table[10n^2-7n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,4,27},50] (* Harvey P. Dale, Apr 06 2020 *)

a(n)=10*n^2-7*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (2*n-1)*(5*n-1).
a(n) = A033571(n) - A008596(n) = A010010(n) - A033571(n).
G.f.: (1+x+18*x^2)/(1-x)^3. - Jaume Oliver Lafont, Mar 27 2009
a(n) = a(n-1) + 20*n - 17 (with a(0)=1). - Vincenzo Librandi, Dec 03 2010
Sum_{n>=0} 1/a(n) = 1 + (2*sqrt(1+2/sqrt(5))*Pi - 2*sqrt(5)*log(phi) - 5*log(5) + 8*log(2))/12, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 22 2022

Typo in definition corrected by Reinhard Zumkeller, Dec 03 2009

cp's OEIS Frontend

A033571 a(n) = (2n + 1)(5*n + 1).

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A158187 a(n) = 10*n^2 + 1.

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A158186 a(n) = 10n^2 - 7n + 1.

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cp's OEIS Frontend

A033571 a(n) = (2*n + 1)*(5*n + 1).

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GAP

Magma

Maple

Mathematica

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Sage

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A158187 a(n) = 10*n^2 + 1.

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A158186 a(n) = 10*n^2 - 7*n + 1.

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Mathematica

PARI

Formula

Extensions

A033571 a(n) = (2n + 1)(5*n + 1).

A158186 a(n) = 10n^2 - 7n + 1.